Mykola Shpot

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine
An extension of the massive field theory approach of Parisi to systems with surfaces is presented. This approach provides the opportunity to study the surface critical behavior directly in space dimensions $d<4$ without having to resort to the $\epsilon$ expansion, especially in three dimensions. The method is elaborated for the the semi-infinite $|{\phi}|^4$ $n$-vector model with a boundary term $c_0\int_{\partial V}{\phi}^2$ in the action. To make the theory UV finite in bulk dimensions $d<4$, a renormalization of the surface enhancement $c_0$ is required, apart of the standard mass renormalization; required norma\-lization conditions for the renormalized theory are given. As a result, in addition to the the usual bulk `mass' (the inverse correlation length) $m$, another mass parameter appears in the theory, the renormalized surface enhancement $c$. Thus the surface renormalization factors depend on the renormalized coupling constant $u$ and the ratio $c/m$. The special and ordinary surface transitions correspond to the limits $m\to 0$ with $c/m\to 0$ and $c/m\to\infty$, respectively. The surface critical exponents of the special and ordinary transitions are given to one-loop order in $2\le d<4$ and to two-loop order at $d=3$. The associated second order series expansions are analyzed by Pad\'e-Borel summation techniques. The resulting numerical estimates for the surface critical exponents are in good agreement with available Monte Carlo simulations. This also holds for the surface crossover exponent $\Phi$, for which the values $\Phi (n\!=\!0)\simeq 0.52$ and $\Phi (n\!=\!1)\simeq 0.54$ are obtained, considerably lower than the previous $\epsilon$-expansion estimates.
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